DOCSLIB.ORG

IV. Structure of Abelian Categories

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
IV. Structure of Abelian Categories IV. Structure of Abelian Categories An abelian category C with coproducts is cocomplete 6.3, and is called an A B 3 category. The sum of any family of subobjects of an object exist, and if, in addition, sum distributes over intersection in the sense that ieI ieI where {A;}w is any directed family of subobjects of any object A of C and B is any subobject of A, then C is said to be an A B 5 category (Grothendieck [57]). (An A B 5 category in which sum distributes over arbitrary intersections is called an AB6 category. See Chapter 15 Exercises.) An AB3 categoi:y is AB5 if and only if it has exact direct limits 14.6, or equivalently, the direct limit of any directed family of subobjects is a subobject (and in this case the direct limit is the sum). A Grothendieck category is an A B 5 catepory with a generator. In such a category, every object has an injective hull 14.17. This is needed to prove the main result of Chapter 14, the (Freyd, Lubkin) embedding theorem 14.21, which states that every small abelian category C has an exact, covariant embedding C '"'+ mod-Z. The (Mitchell) full embedding theorem 15.30 states that a small abelian category C has a full exact embedding C'"'+ mod-A, for a suitable ring A. (This also provides an exact embedding C '"'+ mod-Z.) Quotient categories are introduced in Chapter 15. A full sub­ category [I' of an abelian category d is called a Serre class if for every exact sequence 0 -+ X -+ Y -+ Z -+ 0 in d it is true that Y E [I' ¢? X & Z E [1'. Then, there is a category dl[l' on the objects of d, and morphisms are defined in d 1[I' by: Mordl9' (X, Y) = ~ Mord(X', YIY') ,Y', X/X' E [I' The category dl[l' is called a quotient category, and is abelian 15.7. The Serre subcategory [I' is a localizing subcategory provided that the canoni­ cal functor T: d '"'+ dl[l' has a right adjoint S: dl[l' '"'+ d (called the section functor), and the composition d '"'+ d is called the localizing functor. If d is Grothendieck, a necessary and sufficient condition for IV. Structure of Abelian Categories 485 this is that Y' should be a subcategory with arbitrary coproducts, or equivalently, direct limits taken in d of directed families of subobjects of Y' should belong to Y'. In this case, for each object M of d, there is an exact sequence where X, Y E Y', and N is Y'-closed in the sense 15.15. If e denotes the full subcategory of Y'-closed objects, then the localizing functor induces an equivalence 'IjJ(N) : N --+ STN, and there is an equivalence of categories d/Y' "'" e. The morphism u : 1'\1[ ~r N is determined uniquely up to an equivalence of d, and is called the Y'-envelope of M, denoted u.'/': M --+ M.'/'. If d = mod-A, for some ring A, then A.'/' is a ring, and in classical situations 16.8 when the localizing functor is exact, this functor is naturally isomorphic to @A.'/', and there is an equivalence of categories A mod-A/Y' "'" mod-A.'/'. This situation occurs for the localization theorem 16.15 generalizing to Grothendieck categories localization theorems for commutative rings (Chapters 27 and 28). This method also applies to partial right quotient rings 16.9 generalizing the right quotient rings of Chapters 9 and 10, and to the Johnson maximal right quotient rings of right neat rings 16.12. (Cf. also Chapter 19.) A classical aspect of module theory is the relating to a ring A intrinsi­ cally some extrinsic property of mod-A. Thus, 8.12 every module is semisimple if and only if A is semisimple or 8.21, r.gl. dim A = sup {dim I I I c R} + 1 when A is not semisimple; or the characterisation 11.24 of rings with every module flat, as regular rings in which every principal right ideal is generated by an idempotent. These examples can be multiplied at will, for in a sense, this is ring (and module) theory. In this vein, a natural question raised by localizing sUbcategories of mod-A is the intrinsic relation to mod-A. This is solved by Gabriel's theorem 16,} which establishes a mapping e 1-+ y; (e) = {I c A I A/I E obj e} of the class of localizing subcategories of mod-A to the class consisting of sets of all right ideals of A. The set y; (e) is idempotent and topologizing 486 IV. Structure of Abelian Categories (= idemtop) in the sense of 16.3, and conversely, any idemtop set F of right ideals of A corresponds to the localizing subcategory f/ (F) of mod-A consisting of all modules M such that for every x E M the right ideal annAx E F. Moreover, as stated above for Grothendieck categories the quotient category mod-A/f/ is equivalent to the full subcategory C of mod-A consisting of f/-closed modules, or in the terminology of torsion theories taken up in Chapter 16, the subcategory consisting of torsionfree divisible modules. The same correspondence can be used to characterize Serre classes, and classes of closed full subcategories of mod-A 16.2. In these cases, then the f/ (F)-envelope of a module M is denoted M F instead of M !7(F) , and in particular, A F denotes A !7(F) • Chapter 15 contains the Popesco-Gabriel theorem 15.26 which characterizes Grothendieck categories as quotient categories of module type categories. Chapter 16 concludes with a theorem 16.20 of E. A. and C. L. Walker stating that mod-A is determined in a very strong sense by the endo­ morphism ring of any proper generator. This generalizes the known theorems for principal right ideal domains, and quasi-Frobenius rings. (Cf. Chapter 24.) Chapter 14 Grothendieck Categories Grothendieck [57] introduced the notation for abelian categories which follows: AB3. A coproductive, hence cocomplete, abelian category. A B 4. An A B 3 category in which every coproduct lliEIAi -+ lliElBi of monics {Ai -+ BiliEI is monic. A B 5. An A B 3 category such that for any directed family {AilieI of subobjects of any object X, and any subobject B, ABX*, the dual of ABX, X = 3,4,5. For any totally ordered family {AdiEI of subobjects of any object X, the sum is called union, and is denoted by UiEI Ai' If the family is finite, and consists of Al < ... < An, then UWAi = An. .
Load more

Recommended publications
  • Arxiv:1403.7027V2 [Math.AG] 21 Oct 2015 Ytnoigit Iebnlson Bundles Line Into Tensoring by N Ytednsyfoundation
    ON EQUIVARIANT TRIANGULATED CATEGORIES ALEXEY ELAGIN Abstract. Consider a finite group G acting on a triangulated category T . In this paper we investigate triangulated structure on the category T G of G-equivariant objects in T . We prove (under some technical conditions) that such structure exists. Supposed that an action on T is induced by a DG-action on some DG-enhancement of T , we construct a DG-enhancement of T G. Also, we show that the relation “to be an equivariant category with respect to a finite abelian group action” is symmetric on idempotent complete additive categories. 1. Introduction Triangulated categories became very popular in algebra, geometry and topology in last decades. In algebraic geometry, they arise as derived categories of coherent sheaves on algebraic varieties or stacks. It turned out that some geometry of varieties can be under- stood well through their derived categories and homological algebra of these categories. Therefore it is always interesting and important to understand how different geometrical operations, constructions, relations look like on the derived category side. In this paper we are interested in autoequivalences of derived categories or, more gen- eral, in group actions on triangulated categories. For X an algebraic variety, there are “expected” autoequivalences of Db(coh(X)) which are induced by automorphisms of X or by tensoring into line bundles on X. If X is a smooth Fano or if KX is ample, essentially that is all: Bondal and Orlov have shown in [6] that for smooth irreducible projective b variety X with KX or −KX ample all autoequivalences of D (coh(X)) are generated by automorphisms of X, twists into line bundles on X and translations.
    [Show full text]
  • Classification of Subcategories in Abelian Categories and Triangulated Categories
    CLASSIFICATION OF SUBCATEGORIES IN ABELIAN CATEGORIES AND TRIANGULATED CATEGORIES ATHESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS UNIVERSITY OF REGINA By Yong Liu Regina, Saskatchewan September 2016 c Copyright 2016: Yong Liu UNIVERSITY OF REGINA FACULTY OF GRADUATE STUDIES AND RESEARCH SUPERVISORY AND EXAMINING COMMITTEE Yong Liu, candidate for the degree of Doctor of Philosophy in Mathematics, has presented a thesis titled, Classification of Subcategories in Abelian Categories and Triangulated Categories, in an oral examination held on September 8, 2016. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: Dr. Henning Krause, University of Bielefeld Supervisor: Dr. Donald Stanley, Department of Mathematics and Statistics Committee Member: Dr. Allen Herman, Department of Mathematics and Statistics Committee Member: *Dr. Fernando Szechtman, Department of Mathematics and Statistics Committee Member: Dr. Yiyu Yao, Department of Computer Science Chair of Defense: Dr. Renata Raina-Fulton, Department of Chemistry and Biochemistry *Not present at defense Abstract Two approaches for classifying subcategories of a category are given. We examine the class of Serre subcategories in an abelian category as our first target, using the concepts of monoform objects and the associated atom spectrum [13]. Then we generalize this idea to give a classification of nullity classes in an abelian category, using premonoform objects instead to form a new spectrum so that there is a bijection between the collection of nullity classes and that of closed and extension closed subsets of the spectrum.
    [Show full text]
  • Non-Exactness of Direct Products of Quasi-Coherent Sheaves
    Documenta Math. 2037 Non-Exactness of Direct Products of Quasi-Coherent Sheaves Ryo Kanda Received: March 19, 2019 Revised: September 4, 2019 Communicated by Henning Krause Abstract. For a noetherian scheme that has an ample family of invertible sheaves, we prove that direct products in the category of quasi-coherent sheaves are not exact unless the scheme is affine. This result can especially be applied to all quasi-projective schemes over commutative noetherian rings. The main tools of the proof are the Gabriel-Popescu embedding and Roos’ characterization of Grothendieck categories satisfying Ab6 and Ab4*. 2010 Mathematics Subject Classification: 14F05 (Primary), 18E20, 16D90, 16W50, 13C60 (Secondary) Keywords and Phrases: Quasi-coherent sheaf, divisorial scheme, invertible sheaf, direct product, Gabriel-Popescu embedding, Grothendieck category Contents 1 Introduction 2038 Acknowledgments 2039 2 Gabriel-Popescu embedding and Roos’ theorem 2039 2.1 Preliminaries ............................2039 2.2 Gabriel-Popescu embedding ....................2043 2.3 Roos’ theorem ...........................2045 3 Divisorial noetherian schemes 2047 References 2054 Documenta Mathematica 24 (2019) 2037–2056 2038 Ryo Kanda 1 Introduction The class of Grothendieck categories is a large framework that includes • the category Mod R of right modules over a ring R, • the category QCoh X of quasi-coherent sheaves on a scheme X, and • the category of sheaves of abelian groups on a topological space. One of the significant properties of Mod R for rings R among Grothendieck categories is the exactness of direct products, which is known as Grothendieck’s condition Ab4*. This is immediately verified by direct computation, but it is also a consequence of the fact that Mod R has enough projectives.
    [Show full text]
  • N-Quasi-Abelian Categories Vs N-Tilting Torsion Pairs 3
    N-QUASI-ABELIAN CATEGORIES VS N-TILTING TORSION PAIRS WITH AN APPLICATION TO FLOPS OF HIGHER RELATIVE DIMENSION LUISA FIOROT Abstract. It is a well established fact that the notions of quasi-abelian cate- gories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of t-structures. Firstly, we extend this picture into a hierarchy of n-quasi-abelian categories and n-tilting torsion classes. We prove that any n-quasi-abelian category E admits a “derived” category D(E) endowed with a n-tilting pair of t-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these t-structures as quotient categories of coherent functors, generalizing Auslander’s Formula. Thirdly, we apply our results to Bridgeland’s theory of perverse coherent sheaves for flop contractions. In Bridgeland’s work, the relative dimension 1 assumption guaranteed that f∗-acyclic coherent sheaves form a 1-tilting torsion class, whose associated heart is derived equivalent to D(Y ). We generalize this theorem to relative dimension 2. Contents Introduction 1 1. 1-tilting torsion classes 3 2. n-Tilting Theorem 7 3. 2-tilting torsion classes 9 4. Effaceable functors 14 5. n-coherent categories 17 6. n-tilting torsion classes for n> 2 18 7. Perverse coherent sheaves 28 8. Comparison between n-abelian and n + 1-quasi-abelian categories 32 Appendix A. Maximal Quillen exact structure 33 Appendix B. Freyd categories and coherent functors 34 Appendix C. t-structures 37 References 39 arXiv:1602.08253v3 [math.RT] 28 Dec 2019 Introduction In [6, 3.3.1] Beilinson, Bernstein and Deligne introduced the notion of a t- structure obtained by tilting the natural one on D(A) (derived category of an abelian category A) with respect to a torsion pair (X , Y).
    [Show full text]
  • Basic Category Theory and Topos Theory
    Basic Category Theory and Topos Theory Jaap van Oosten Jaap van Oosten Department of Mathematics Utrecht University The Netherlands Revised, February 2016 Contents 1 Categories and Functors 1 1.1 Definitions and examples . 1 1.2 Some special objects and arrows . 5 2 Natural transformations 8 2.1 The Yoneda lemma . 8 2.2 Examples of natural transformations . 11 2.3 Equivalence of categories; an example . 13 3 (Co)cones and (Co)limits 16 3.1 Limits . 16 3.2 Limits by products and equalizers . 23 3.3 Complete Categories . 24 3.4 Colimits . 25 4 A little piece of categorical logic 28 4.1 Regular categories and subobjects . 28 4.2 The logic of regular categories . 34 4.3 The language L(C) and theory T (C) associated to a regular cat- egory C ................................ 39 4.4 The category C(T ) associated to a theory T : Completeness Theorem 41 4.5 Example of a regular category . 44 5 Adjunctions 47 5.1 Adjoint functors . 47 5.2 Expressing (co)completeness by existence of adjoints; preserva- tion of (co)limits by adjoint functors . 52 6 Monads and Algebras 56 6.1 Algebras for a monad . 57 6.2 T -Algebras at least as complete as D . 61 6.3 The Kleisli category of a monad . 62 7 Cartesian closed categories and the λ-calculus 64 7.1 Cartesian closed categories (ccc's); examples and basic facts . 64 7.2 Typed λ-calculus and cartesian closed categories . 68 7.3 Representation of primitive recursive functions in ccc's with nat- ural numbers object .
    [Show full text]
  • Weak Subobjects and Weak Limits in Categories and Homotopy Categories Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 38, No 4 (1997), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES MARCO GRANDIS Weak subobjects and weak limits in categories and homotopy categories Cahiers de topologie et géométrie différentielle catégoriques, tome 38, no 4 (1997), p. 301-326 <http://www.numdam.org/item?id=CTGDC_1997__38_4_301_0> © Andrée C. Ehresmann et les auteurs, 1997, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET Volume XXXVIII-4 (1997) GEOMETRIE DIFFERENTIELLE CATEGORIQUES WEAK SUBOBJECTS AND WEAK LIMITS IN CATEGORIES AND HOMOTOPY CATEGORIES by Marco GRANDIS R6sumi. Dans une cat6gorie donn6e, un sousobjet faible, ou variation, d’un objet A est defini comme une classe d’6quivalence de morphismes A valeurs dans A, de faqon a étendre la notion usuelle de sousobjet. Les sousobjets faibles sont lies aux limites faibles, comme les sousobjets aux limites; et ils peuvent 8tre consid6r6s comme remplaqant les sousobjets dans les categories "a limites faibles", notamment la cat6gorie d’homotopie HoTop des espaces topologiques, ou il forment un treillis de types de fibration sur 1’espace donn6. La classification des variations des groupes et des groupes ab£liens est un outil important pour d6terminer ces types de fibration, par les foncteurs d’homotopie et homologie. Introduction We introduce here the notion of weak subobject in a category, as an extension of the notion of subobject.
    [Show full text]
  • Introduction to Category Theory (Notes for Course Taught at HUJI, Fall 2020-2021) (UNPOLISHED DRAFT)
    Introduction to category theory (notes for course taught at HUJI, Fall 2020-2021) (UNPOLISHED DRAFT) Alexander Yom Din February 10, 2021 It is never true that two substances are entirely alike, differing only in being two rather than one1. G. W. Leibniz, Discourse on metaphysics 1This can be imagined to be related to at least two of our themes: the imperative of considering a contractible groupoid of objects as an one single object, and also the ideology around Yoneda's lemma ("no two different things have all their properties being exactly the same"). 1 Contents 1 The basic language 3 1.1 Categories . .3 1.2 Functors . .7 1.3 Natural transformations . .9 2 Equivalence of categories 11 2.1 Contractible groupoids . 11 2.2 Fibers . 12 2.3 Fibers and fully faithfulness . 12 2.4 A lemma on fully faithfulness in families . 13 2.5 Definition of equivalence of categories . 14 2.6 Simple examples of equivalence of categories . 17 2.7 Theory of the fundamental groupoid and covering spaces . 18 2.8 Affine algebraic varieties . 23 2.9 The Gelfand transform . 26 2.10 Galois theory . 27 3 Yoneda's lemma, representing objects, limits 27 3.1 Yoneda's lemma . 27 3.2 Representing objects . 29 3.3 The definition of a limit . 33 3.4 Examples of limits . 34 3.5 Dualizing everything . 39 3.6 Examples of colimits . 39 3.7 General colimits in terms of special ones . 41 4 Adjoint functors 42 4.1 Bifunctors . 42 4.2 The definition of adjoint functors . 43 4.3 Some examples of adjoint functors .
    [Show full text]
  • On the Voevodsky Motive of the Moduli Space of Higgs Bundles on a Curve
    ON THE VOEVODSKY MOTIVE OF THE MODULI SPACE OF HIGGS BUNDLES ON A CURVE VICTORIA HOSKINS AND SIMON PEPIN LEHALLEUR Abstract We study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky's triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic. Contents 1. Introduction 1 1.1. Moduli of Higgs bundles and their cohomological invariants 1 1.2. Our results 2 2. Moduli of Higgs bundles and moduli of chains 4 2.1. Moduli of Higgs bundles 5 2.2. The scaling action on the moduli space of Higgs bundles 5 2.3. Moduli of chains 6 2.4. Variation of stability and Harder{Narasimhan stratification results 8 3. Voevodsky's category of effective motives and motives of stacks 8 3.1. Motives of schemes 8 3.2. Motives of smooth exhaustive stacks 9 4. Motivic non-abelian Hodge correspondence 10 5. Hecke correspondences and the stack of generically surjective chains 11 5.1. Motives of stacks of Hecke correspondences 11 5.2. Motive of the stack of generically surjective chains 12 6. The recursive description of the motive of the Higgs moduli space 13 6.1. Motivic consequences of wall-crossing and HN recursions 13 6.2.
    [Show full text]
  • Noncommutative Stacks
    Noncommutative Stacks Introduction One of the purposes of this work is to introduce a noncommutative analogue of Artin’s and Deligne-Mumford algebraic stacks in the most natural and sufficiently general way. We start with quasi-coherent modules on fibered categories, then define stacks and prestacks. We define formally smooth, formally unramified, and formally ´etale cartesian functors. This provides us with enough tools to extend to stacks the glueing formalism we developed in [KR3] for presheaves and sheaves of sets. Quasi-coherent presheaves and sheaves on a fibered category. Quasi-coherent sheaves on geometric (i.e. locally ringed topological) spaces were in- troduced in fifties. The notion of quasi-coherent modules was extended in an obvious way to ringed sites and toposes at the moment the latter appeared (in SGA), but it was not used much in this generality. Recently, the subject was revisited by D. Orlov in his work on quasi-coherent sheaves in commutative an noncommutative geometry [Or] and by G. Laumon an L. Moret-Bailly in their book on algebraic stacks [LM-B]. Slightly generalizing [R4], we associate with any functor F (regarded as a category over a category) the category of ’quasi-coherent presheaves’ on F (otherwise called ’quasi- coherent presheaves of modules’ or simply ’quasi-coherent modules’) and study some basic properties of this correspondence in the case when the functor defines a fibered category. Imitating [Gir], we define the quasi-topology of 1-descent (or simply ’descent’) and the quasi-topology of 2-descent (or ’effective descent’) on the base of a fibered category (i.e.
    [Show full text]
  • Essential Properties of Pro-Objects in Grothendieck Categories Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 20, No 1 (1979), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES TIMOTHY PORTER Essential properties of pro-objects in Grothendieck categories Cahiers de topologie et géométrie différentielle catégoriques, tome 20, no 1 (1979), p. 3-57 <http://www.numdam.org/item?id=CTGDC_1979__20_1_3_0> © Andrée C. Ehresmann et les auteurs, 1979, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE Vol. XX -1 ( 1979 ) ET GEOMETRIE DIFFERENTIELLE ESSENTIAL PROPERTIES OF PRO-OBJECTS IN GROTHENDIECK CATEGORIES by Timothy PORTER There is a class of problems in Algebra, algebraic Topology and category Theory which can be subsummed under the question : When does a «structure &#x3E;&#x3E; on a category C extend to a similar « structure» on the cor- responding procategory pro ( C ) ? Thus one has the work of Edwards and Hastings [7] and the author [27,28,29] on extending a «model category for homotopy theory » a la Quillen [35] from a category C , usually the category of simplicial sets or of chain complexes over some ring, to give a homotopy theory or homo- topical algebra in pro(C) . ( It is worth noting that the two approaches differ considerably, but they agree in the formation of the homotopy categ- ory / category of fractions Hopro(C) which, in this case, is distinct from the prohomotopy category&#x3E;&#x3E; proHo ( C ) .
    [Show full text]
  • Groups and Categories
    \chap04" 2009/2/27 i i page 65 i i 4 GROUPS AND CATEGORIES This chapter is devoted to some of the various connections between groups and categories. If you already know the basic group theory covered here, then this will give you some insight into the categorical constructions we have learned so far; and if you do not know it yet, then you will learn it now as an application of category theory. We will focus on three different aspects of the relationship between categories and groups: 1. groups in a category, 2. the category of groups, 3. groups as categories. 4.1 Groups in a category As we have already seen, the notion of a group arises as an abstraction of the automorphisms of an object. In a specific, concrete case, a group G may thus consist of certain arrows g : X ! X for some object X in a category C, G ⊆ HomC(X; X) But the abstract group concept can also be described directly as an object in a category, equipped with a certain structure. This more subtle notion of a \group in a category" also proves to be quite useful. Let C be a category with finite products. The notion of a group in C essentially generalizes the usual notion of a group in Sets. Definition 4.1. A group in C consists of objects and arrows as so: m i G × G - G G 6 u 1 i i i i \chap04" 2009/2/27 i i page 66 66 GROUPSANDCATEGORIES i i satisfying the following conditions: 1.
    [Show full text]
  • Arxiv:Math/9909030V1 [Math.CT] 5 Sep 1999 B.A3ctgr,I Hc O N Ietdfamily Directed Any for Which in Ab3-Category, a Equivalently, Ab5
    GROTHENDIECK CATEGORIES GRIGORY GARKUSHA Contents 0. Introduction 1 1. Preliminaries 6 1.1. Ab-conditions 6 1.2. Localization in Grothendieck categories 7 1.3. Lattices of localizing subcategories 10 1.4. Locally finitely presented Grothendieck categories 11 2. Grothendieck categories possessing finitely generated projective generators 13 2.1. The Gabriel topology 14 2.2. Localization in module categories 17 3. The structure of localizing subcategories 20 3.1. Negligible objects and covering morphisms 22 4. Representation of Grothendieck categories 26 4.1. Projective generating sets 32 5. Finiteness conditions for localizing subcategories 36 5.1. Left exact functors 40 5.2. The Ziegler topology 43 6. Categories of generalized modules 45 6.1. Tensor products 45 6.2. Auslander-Gruson-Jensen Duality 48 7. Grothendieck categories as quotient categories of (mod Aop, Ab) 50 8. Absolutely pure and flat modules 54 References 60 arXiv:math/9909030v1 [math.CT] 5 Sep 1999 0. Introduction The theory has its origin in the work of Grothendieck [Grk] who introduced the following notation of properties of abelian categories: Ab3. An abelian category with coproducts or equivalently, a cocomplete abelian cate- gory. Ab5. Ab3-category, in which for any directed family {Ai}i∈I of subobjects of an arbi- trary object X and for any subobject B of X the following relation holds: (X Ai) ∩ B = X(Ai ∩ B) i∈I i∈I 1991 Mathematics Subject Classification. Primary 18E, 16D. 2 GRIGORY GARKUSHA Ab5-categories possessing a family of generators are called Grothendieck categories. They constitute a natural extension of the class of module categories, with which they share a great number of important properties.
    [Show full text]